L(s) = 1 | − 2-s − 4-s + 5-s − 2·7-s + 3·8-s − 10-s + 2·11-s − 4·13-s + 2·14-s − 16-s − 2·17-s − 19-s − 20-s − 2·22-s + 4·23-s + 25-s + 4·26-s + 2·28-s − 4·29-s − 5·32-s + 2·34-s − 2·35-s + 38-s + 3·40-s − 10·43-s − 2·44-s − 4·46-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s − 0.755·7-s + 1.06·8-s − 0.316·10-s + 0.603·11-s − 1.10·13-s + 0.534·14-s − 1/4·16-s − 0.485·17-s − 0.229·19-s − 0.223·20-s − 0.426·22-s + 0.834·23-s + 1/5·25-s + 0.784·26-s + 0.377·28-s − 0.742·29-s − 0.883·32-s + 0.342·34-s − 0.338·35-s + 0.162·38-s + 0.474·40-s − 1.52·43-s − 0.301·44-s − 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.612126256967840032330072250055, −9.149810416408305523281784925699, −8.255821268467971823002210114232, −7.20655519160802512316474001684, −6.50028762370085906668100412134, −5.26580762429388022199308033556, −4.39966598748280163950451872807, −3.13810729229125509539196623139, −1.68584050985555553185267943448, 0,
1.68584050985555553185267943448, 3.13810729229125509539196623139, 4.39966598748280163950451872807, 5.26580762429388022199308033556, 6.50028762370085906668100412134, 7.20655519160802512316474001684, 8.255821268467971823002210114232, 9.149810416408305523281784925699, 9.612126256967840032330072250055